# What do we mean by 'Equivalent Projective representation ?

by S_klogW
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 Mentor P: 18,036 I've never done anything about projective representations before, so this post is just a guess. But it would make sense to define first $$Z=\{cI_n~\vert~c\in \mathbb{R}\}$$ Then we define a projective representation as a group homomorphism $$\rho: G\rightarrow GL_n(\mathbb{R})/Z$$ This last group is often called $PGL_n(\mathbb{R})$, or the projective general linear group. Given, $\rho,\rho^\prime$ projective representations, it would make sense to define them equivalent if there exist $U\in O_n(\mathbb{R})/Z$ such that $$\rho(g)=U\cdot \rho^\prime(g)\cdot U^{-1}$$ for all $g\in G$. The complex case is similar.